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# hw7 - θ(d Deduce that Z 35 ∼ = Z 5 × Z 7(3 points 5 Let...

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Math 3124 Tuesday, October 18 Seventh Homework Due 12:30 p.m., Tuesday October 25 1. Explain why S 5 × S 4 S 6 × Z 4 . (Hint: 15) (2 points) 2. Problem 21.6(a) on page 109. (1 point) 3. Problem 21.27 on page 110. Hint: The right cosets of N in G are N and G \ N , and similarly the left cosets of N in G are N and G \ N . (2 points) 4. Define θ : Z 35 Z 5 × Z 7 by θ ([ a ]) = ([ a ] , [ a ]) . (a) Show that θ is well-defined. (b) Show that θ is a homomorphism. (c) What is ker
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Unformatted text preview: θ ? (d) Deduce that Z 35 ∼ = Z 5 × Z 7 . (3 points) 5. Let G be a group and let θ : S 3 → G be a homomorphism such that θ (( 1 2 )) = e G . (a) Prove that h ( 1 2 ) i ⊆ ker θ and h ( 1 2 ) i 6 S 3 . (b) Determine ker θ (why is ker θ 6 = h ( 1 2 ) i ?) (c) Deduce that θ (( 1 3 )) = e G . (3 points) (5 problems, 11 points altogether)...
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